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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaovcang 6201* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcand 6202* Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcanrd 6203* Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovcan 6204* Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)

Theoremcaovordig 6205* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremcaovordid 6206* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremcaovordg 6207* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaovordd 6208* Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord2d 6209* Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord3d 6210* Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovord 6211* Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)

Theoremcaovord2 6212* Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)

Theoremcaovord3 6213* Ordering law. (Contributed by NM, 29-Feb-1996.)

Theoremcaovdig 6214* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)

Theoremcaovdid 6215* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdir2d 6216* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdirg 6217* Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)

Theoremcaovdird 6218* Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Theoremcaovdi 6219* Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)

Theoremcaov32d 6220* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov12d 6221* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov31d 6222* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov13d 6223* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov4d 6224* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov411d 6225* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov42d 6226* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremcaov32 6227* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov12 6228* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov31 6229* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov13 6230* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov4 6231* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov411 6232* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaov42 6233* Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Theoremcaovdir 6234* Reverse distributive law. (Contributed by NM, 26-Aug-1995.)

Theoremcaovdilem 6235* Lemma used by real number construction. (Contributed by NM, 26-Aug-1995.)

Theoremcaovlem2 6236* Lemma used in real number construction. (Contributed by NM, 26-Aug-1995.)

Theoremcaovmo 6237* Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)

Theoremgrprinvlem 6238* Lemma for grprinvd 6239. (Contributed by NM, 9-Aug-2013.)

Theoremgrprinvd 6239* Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Theoremgrpridd 6240* Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

2.4.11  "Maps to" notation

Theoremelmpt2cl 6241* If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremelmpt2cl1 6242* If a two-parameter class is not empty, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremelmpt2cl2 6243* If a two-parameter class is not empty, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremelovmpt2 6244* Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 14947, islmhm 16044. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremrelmptopab 6245* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremf1ocnvd 6246* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremf1od 6247* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremf1ocnv2d 6248* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremf1o2d 6249* Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

TheoremxpexgALT 6250 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4943 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremf1opw2 6251* A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6252 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)

Theoremf1opw 6252* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremsuppss2 6253* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremsuppssfv 6254* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremsuppssov1 6255* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

2.4.12  Function operation

Syntaxcof 6256 Extend class notation to include mapping of an operation to a function operation.

Syntaxcofr 6257 Extend class notation to include mapping of a binary relation to a function relation.

Definitiondf-of 6258* Define the function operation map. The definition is designed so that if is a binary operation, then is the analogous operation on functions which corresponds to applying pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)

Definitiondf-ofr 6259* Define the function relation map. The definition is designed so that if is a binary relation, then is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofeq 6260 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofreq 6261 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofexg 6262 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)

Theoremnfof 6263* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremnfofr 6264* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremoffval 6265* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrfval 6266* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofval 6267 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrval 6268 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremoffn 6269 The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremfnfvof 6270 Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)

Theoremoffval3 6271* General value of with no assumptions on functionality of and . (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremoffres 6272 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremoff 6273* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofres 6274 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremoffval2 6275* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrfval2 6276* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofco 6277 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremoffveq 6278* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremoffveqb 6279* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremofc1 6280 Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofc2 6281 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)

Theoremofc12 6282 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofref 6283* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofinvl 6284* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)

Theoremcaofid0l 6285* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Theoremcaofid0r 6286* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Theoremcaofid1 6287* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofid2 6288* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofcom 6289* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaofrss 6290* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofass 6291* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaoftrn 6292* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofdi 6293* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaofdir 6294* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)

Theoremcaonncan 6295* Transfer nncan 9276-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Theoremofmres 6296* Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 6298, allowing it to be used as a function or structure argument. By ofmresval 6297, the restricted operation map values are the same as the original values, allowing theorems for to be reused. (Contributed by NM, 20-Oct-2014.)

Theoremofmresval 6297 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremofmresex 6298 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremsuppssof1 6299* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

2.4.13  First and second members of an ordered pair

Syntaxc1st 6300 Extend the definition of a class to include the first member an ordered pair function.

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